Quantum Walk Search on Kronecker Graphs

TL;DR

This paper investigates quantum search algorithms on Kronecker graphs, demonstrating analytically and numerically that they support optimal search times proportional to the square root of the network size, similar to Grover's algorithm.

Contribution

It provides the first analytical proofs of optimal quantum search on Kronecker graphs with complete initiators, extending to higher orders through numerical simulations.

Findings

Quantum search on first-, second-, and third-order Kronecker graphs is optimal with O(√N) time.

Numerical results suggest higher-order Kronecker graphs also support optimal quantum search.

Analytical proofs are provided for the initial orders of Kronecker graphs.

Abstract

Kronecker graphs, obtained by repeatedly performing the Kronecker product of the adjacency matrix of an "initiator" graph with itself, have risen in popularity in network science due to their ability to generate complex networks with real-world properties. In this paper, we explore spatial search by continuous-time quantum walk on Kronecker graphs. Specifically, we give analytical proofs for quantum search on first-, second-, and third-order Kronecker graphs with the complete graph as the initiator, showing that search takes Grover's $O(\sqrt{N})$ time. Numerical simulations indicate that higher-order Kronecker graphs with the complete initiator also support optimal quantum search.